Online Learning for Equilibrium Pricing in Markets under Incomplete Information

TL;DR

This paper develops online algorithms for market equilibrium pricing under incomplete information, achieving near-optimal regret bounds in fixed and varying cost scenarios, with contextual information improving performance.

Contribution

It introduces novel online learning algorithms for equilibrium pricing with incomplete information, including methods that adapt to changing costs using contextual hints.

Findings

Algorithms achieve $O(1)$ regret for constant demand and $O(\sqrt{T})$ for variable demand with fixed costs.

In general, no sublinear regret algorithms exist for varying costs without additional information.

Contextual information enables sublinear regret algorithms in dynamic cost settings.

Abstract

The computation of equilibrium prices at which the supply of goods matches their demand typically relies on complete information on agents' private attributes, e.g., suppliers' cost functions, which are often unavailable in practice. Motivated by this practical consideration, we consider the problem of learning equilibrium prices over a horizon of $T$ periods in the incomplete information setting wherein a market operator seeks to satisfy the customer demand for a commodity by purchasing it from competing suppliers with cost functions unknown to the operator. We first consider the setting when suppliers' cost functions are fixed and develop algorithms that, on three pertinent regret metrics, simultaneously achieve a regret of $O(1)$ when the customer demand is constant over time, and $O(\sqrt{T})$ when the demand varies over time. In the setting when the suppliers' cost functions vary over time, we demonstrate that, in general, no online algorithm can achieve sublinear regret on all three metrics. Thus, we consider an augmented setting wherein the operator has access to hints/contexts that reflect the variation in the cost functions and propose an algorithm with sublinear regret in this augmented setting. Finally, we present numerical experiments that validate our results and discuss various model extensions.